Soliton pair dynamics in patterned ferromagnetic ellipses

F. Y. Fradin; K. S. Buchanan; K. Yu. Guslienko; M. Grimsditch; P. E. Roy; S. D. Bader; V. Novosad

REVIEW

Soliton pair dynamics in patterned ferromagnetic ellipses

K. S. Buchanan, P. E. Roy, M. Grimsditch, F. Y. Fradin, K. Yu. Guslienko, S. D. Bader, V. Novosad

Materials Science · Submitted 22 Feb 2006 · 34 references

Pith's verdict: First experimental look at coupled vortex-pair translational modes in a single confined ferromagnet; the mode assignment is plausible but would benefit from full-size simulations.

T0 · AI review

How this review was made: An AI referee (deepseek-v4-flash) read the paper on August 28, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

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arXiv IDcond-mat/0602509 (version 1)

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CitationF. Y. Fradin, K. S. Buchanan, K. Yu. Guslienko, M. Grimsditch, P. E. Roy, S. D. Bader, V. Novosad. "Soliton pair dynamics in patterned ferromagnetic ellipses." arXiv:cond-mat/0602509 (2006). https://pith.science/paper/5WU7JYHW

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Editor's note

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You should know this is the first experimental report of dynamic coupling between two magnetic vortices confined in a single ferromagnetic ellipse. The microwave measurements show two sub-GHz resonances (142 and 188 MHz) in the two-vortex state, and the authors assign them to translational modes of vortex pairs with parallel and antiparallel core polarizations using LLG simulations and a Thiele-model product check. The central idea—polarizations barely matter for static equilibrium but control the dynamics—is clearly argued and plausible.

What is genuinely new: previous work covered single-vortex gyrotropic modes and analytical predictions for vortex pairs, but not the confined two-vortex experimental spectrum. The selection rules (which modes a uniform rf field can excite depending on p1p2 and field direction) are elegant and provide a discriminator that is largely independent of absolute frequency calibration. The product invariance of the Thiele-model eigenfrequencies for the two polarization combinations is a nice parameter-free consistency check, and the two products agree to better than 1% in simulation.

The soft spots are real but not disqualifying. The simulations were done on half-size ellipses (1.5 by 0.75 um, 20 nm) rather than the experimental 3 by 1.5 um, 40 nm dots, and the text admits simulated frequencies run up to 10% higher, with the scaling tests not shown. Since the two measured peaks are only about 46 MHz apart, a 10% shift in a simulated branch could in principle swap the assignment. However, the assignments are not based on absolute frequency alone: the mode excited by both rf directions (142 MHz) naturally corresponds to the p1p2=+1 branch, and the 188 MHz peak appears only for one rf direction, matching the p1p2=-1 selection rule. The field-dependent splitting and symmetry also line up. So the stress-test concern is legitimate, but I think the paper's inferences survive; they would be much more secure with full-size simulations or a side-by-side table of measured and scaled simulated frequencies. The missing lower branch for one p1p2=-1 mode is a gap, but the authors acknowledge it, and it is not fatal to the main claim.

The static polarization independence rests only on simulation; there is no direct experimental measurement of core positions versus polarization. That is a limitation, but the static result is not the headline. The dynamic result is.

Who should read this: people working on vortex dynamics, confined solitons, or nanomagnet resonance. It deserves a serious referee. I would send it out. If it were mine to decide, I would ask the authors to add a full-size simulation comparison or at least quantify the scaling error branch-by-branch, but I would not reject on the current evidence.

Referee report

Major objections (2)

[Micromagnetic simulations and discussion of Fig. 4] The identification of the measured 142 and 188 MHz peaks with specific polarization states rests on LLG simulations performed for a 1.5 by 0.75 micrometer, 20 nm thick ellipse, exactly half the size of the experimental 3 by 1.5 micrometer, 40 nm ellipses. The manuscript states that the simulated frequencies are uniformly higher and that 'tests for selected cases' show a scaling error of up to 10%, but those tests are not described or shown, and no full-size simulation is presented for calibration. Because the two experimental peaks are separated by only about 46 MHz, a 10% correction (approximately 15-19 MHz at these frequencies) is comparable to the inter-peak spacing and to the spacing between the branches in Fig. 4, so the mode-by-mode assignment is not uniquely established. In addition, the lower branch of the p1p2 = -1 pair and the H-along-x resonance are not resolved experimentally, which removes two of the predictions from the comparison. The authors should report the scaling tests, provide full-size simulation results for at least a few field values, or otherwise quantify the mapping from simulated to measured branches.
[Static properties and Fig. 1] The conclusion that vortex polarization has a negligible effect on static properties is a headline finding, but the only support in the manuscript is the sentence that the simulations lead to the same energies and core positions independent of the polarizations. No numerical values are given, so the reader cannot judge how 'negligible' the effect is, or whether it would be resolvable in the experiment. Because the interpretation of the two resonant peaks as p1p2 = +1 and p1p2 = -1 modes presumes that both polarization states are present in the ensemble with comparable probability, the authors should provide the computed energy difference (or an upper bound) and the core-displacement difference between the two polarization configurations.

Minor objections (5)

[Fig. 2] The symbols in Fig. 2b are described only in the caption; a legend inside the figure or explicit labels on the branches would make the field-dependent data much easier to follow.
[Thiele model section] The equations defining the energy function W(X1,X2) and the stiffness coefficients contain typesetting errors and incomplete expressions; these should be corrected so that the model can be followed without reference to the original Thiele paper.
[Experimental results, paragraph beginning 'A perturbation along y...'] The sentence describing the additional line at 80 MHz states that it was observed 'for a perturbation along y' and then 'for a perturbation along x'; the relative orientation of the static and rf fields should be restated unambiguously for this feature.
[Chirality population assumption, discussion of Fig. 3b] The assumption that equal numbers of ellipses have central-region magnetization parallel and antiparallel to the applied y-field is plausible but not experimentally verified; a brief comment on whether MFM or other data constrain this population would be useful.
[References] Several references have incomplete bibliographic details (e.g., refs. 20, 22, 30, 34), and one inline citation to the LLG Micromagnetics Simulator appears as a URL rather than a formatted reference; these should be corrected to journal style.

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The paper

The paper claims

The paper reports the first experimental evidence that two magnetic vortices confined in one Permalloy ellipse move as a coupled pair, with the relative core polarization determining which translational modes a microwave field excites.

The reading

This paper reports the first experimental evidence that two magnetic vortices confined in a single ferromagnetic ellipse move as a coupled pair, not as independent objects. Microwave reflection measurements on arrays of lithographically patterned Permalloy ellipses show strong sub-GHz resonances near 142 and 188 MHz that appear only in the two-vortex state. Comparing with micromagnetic simulations and a two-vortex model based on the gyroforce, the resonances are assigned to the translational modes of vortex pairs with parallel ($p_1p_2 = +1$) or antiparallel ($p_1p_2 = -1$) core polarizations. The key message is that core polarization barely affects the static positions and energies of the two vortices but completely determines which dynamic modes a uniform microwave field can excite. If the assignment is right, this is the first experimental characterization of confined vortex-pair dynamics and a template for studying soliton-soliton interactions in nanomagnets.

What carries the argument

The load-bearing object is the coupled equation of motion for the two vortex-core positions, built from the gyrotropic force (a force perpendicular to core velocity, proportional to core polarization $p_j$ and to the gyrovector $G$) balanced against restoring forces derived from an energy $W(\mathbf{X}_1, \mathbf{X}_2)$. For the elliptical dot the relevant coefficients are the self-stiffnesses $\kappa_x$, $\kappa_y$ and the vortex-vortex couplings $\mu_x$, $\mu_y$; the sign of $p_j$ enters only through the gyroforce, which is why static properties are polarization-independent while the dynamics are not. The mode classification $(i,i)$, $(o,o)$, $(i,o)$, $(o,i)$ follows from the relative phase of the two cores' elliptical orbits, and micromagnetic simulations supply the concrete frequency branches that are matched to the measurements.

Extended reading notes

Core claim

The paper's central claim is that two magnetic vortices confined in a single elliptical Permalloy dot form a coupled dynamical system whose excitation spectrum is governed by the relative sign of the core polarizations. The two experimentally observed sub-GHz resonances, near 142 and 188 MHz at zero field, are assigned to the translational (gyrotropic) modes of the vortex pair; simulations of a half-size ellipse show that for $p_1p_2 = +1$ the same mode is excited by either in-plane field direction, while for $p_1p_2 = -1$ the two field directions excite different modes. The paper also establishes that the vortex polarizations have virtually no effect on the static energy or core equilibrium positions, but they change the sense of gyration of each core and therefore control the dynamics. A phenomenological model based on the gyroforce and a harmonic two-core energy landscape yields two eigenfrequency pairs whose product is the same for both polarization combinations, and this invariance is confirmed by the micromagnetic simulations at zero and non-zero fields.

Load-bearing premise

The mode-by-mode assignment relies on scaling the experimental 3 by 1.5 micrometer, 40 nm thick ellipses down to half-size, 20 nm thick simulation cells; if the roughly 10 percent frequency offset and the unresolved lower $p_1p_2 = -1$ branch hide a different mode ordering, the identification of the observed lines would fail.

Editorial extensions

If this is right

The 142 and 188 MHz absorption lines provide a non-imaging fingerprint of the two-vortex state, clearly separated from the lower-frequency single-vortex resonance.
For $p_1p_2 = +1$ the same translational mode responds to microwave fields along either ellipse axis, whereas for $p_1p_2 = -1$ the excited mode and its frequency depend on the field direction, so microwave absorption can distinguish the two polarization configurations.
A static field along the short axis splits the modes, and the splitting pattern reflects the mixture of vortex chiralities in the ensemble; fields along the long axis leave the frequencies nearly unchanged.
The product of the two eigenfrequencies for each polarization combination is invariant and agrees with the phenomenological model, giving a quantitative consistency check that survives away from zero field.
Three of the four predicted translational modes are excitable by a spatially uniform r.f. field; the fourth, the in-phase mode of a $p_1p_2 = +1$ pair, is dark but its existence is confirmed in simulation, so a full four-mode spectrum awaits a non-uniform or local excitation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural next experiment is a single-ellipse or time-resolved imaging measurement that tracks both cores during the 142 and 188 MHz resonances; if the cores rotate in the same sense at one frequency and opposite senses at the other, the polarization-dependent assignment is directly confirmed.
Because the in-phase $p_1p_2 = +1$ mode is not excitable by a uniform field, a local excitation source could reveal it and complete the four-mode spectrum without relying on scaled-down simulations.
The same gyroforce-plus-stiffness reasoning should extend to other confined two-soliton systems, such as vortex-antivortex pairs or vortices in neighbouring dots, where the sign of the gyrovector will produce different mode symmetries and selection rules.

Original abstract

Confinement alters the energy landscape of nanoscale magnets, leading to the appearance of unusual magnetic states, such as vortices, for example. Many basic questions concerning dynamical and interaction effects remain unanswered, and nanomagnets are convenient model systems for studying these fundamental physical phenomena. A single vortex in restricted geometry, also known as a non-localized soliton, possesses a characteristic translational excitation mode that corresponds to spiral-like motion of the vortex core around its equilibrium position. Here, we investigate, by a microwave reflection technique, the dynamics of magnetic soliton pairs confined in lithographically defined, ferromagnetic Permalloy ellipses. Through a comparison with micromagnetic simulations, the observed strong resonances in the subgigahertz frequency range can be assigned to the translational modes of vortex pairs with parallel or antiparallel core polarizations. Vortex polarizations play a negligible role in the static interaction between two vortices, but their effect dominates the dynamics.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.15 · human participants 0.01 · controlled comparison 0.10 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.61 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Confinement alters the energy landscape of nanoscale magnets, leading to the appearance of unusual magnetic states, such as vortices, for example. Many basic questions concerning dynamical and interaction effects remain unanswered, and nanomagnets are convenient model systems for studying these fundamental physical phenomena. A single vortex in restricted geometry, also known as a non-localized soliton, possesses a characteristic translational excitation mode that corresponds to spiral-like motion of the vortex core around its equilibrium position. Here, we investigate, by a microwave reflection technique, the dynamics of magnetic soliton pairs confined in lithographically defined, ferromagnetic Permalloy ellipses. Through a comparison with micromagnetic simulations, the observed strong resonances in the subgigahertz frequency range can be assigned to the translational modes of vortex pairs with parallel or antiparallel core polarizations. Vortex polarizations play a negligible role in the static interaction between two vortices, but their effect dominates the dynamics.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Confinement alters the energy landscape of nanoscale magnets, leading to the appearance of unusual magnetic states, such as vortices, for example.	background 1.00	0.05	0.02	asserted 0.99
s1	Many basic questions concerning dynamical and interaction effects remain unanswered, and nanomagnets are convenient model systems for studying these fundamental physical phenomena.	gap 0.82	0.04	0.02	asserted 1.00
s2	A single vortex in restricted geometry, also known as a non-localized soliton, possesses a characteristic translational excitation mode that corresponds to spiral-like motion of the vortex core around its equilibrium position.	background 1.00	0.05	0.04	asserted 1.00
s3	Here, we investigate, by a microwave reflection technique, the dynamics of magnetic soliton pairs confined in lithographically defined, ferromagnetic Permalloy ellipses.	contribution 0.96	0.96	0.05	asserted 0.95
s4	Through a comparison with micromagnetic simulations, the observed strong resonances in the subgigahertz frequency range can be assigned to the translational modes of vortex pairs with parallel or antiparallel core polarizations.	interpretation 0.73	0.94	0.49	asserted 0.98
s5	Vortex polarizations play a negligible role in the static interaction between two vortices, but their effect dominates the dynamics.	result 0.78	0.79	0.53	asserted 1.00

What kind of paper the classifier reports this is: empirical study 1.00 · evidence from observation or experiment 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 2.** Experimentally measured vortex-pair resonance frequencies. a) Representative real (solid) and imaginary (dotted) impedance derivative spectra for H of 10 Oe applied along the long axis of the 3 by 1.5 µm Permalloy ellipses shown in the optical microscope image (inset). Two peaks are observed after first saturating the sample along the long axis. The field dependence of the resonance frequencies is shown in b), where the results for H along the long and short axes are displayed at negative and positive fields, respectively. Solid and open symbols indicate that the r.f. field was…

Showing 1 of the paper's 2 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:cond-mat/0602509 by the authors.

References

Works cited (34)

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Nakazawa, M., Yamada, E. & Kubota, H. Coexistence of self-induced transparen and nolinear Schrodinger soliton. Phys. Rev. Lett. 66, 2625-2628 (1991)

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Matthews, M. R. et al. Vortices in a Bose-Einstein condensate. Phys. Rev. Lett. 83, 2498- 2501 (1999)

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Lamb, H. Hydrodynamics (Dover Publications, New York, 1945)

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Saitoh, E., Miyajima, H., Yamaoka, T. & G. Tatara, G. Current-induced resonance and mass determ

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Wachowiak, A. et al. Direct observation of internal spin structure of magnetic vor cores. Science 298, 577-580 (2002)

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Guslienko, K. Yu., Novosad, V., Otani, Y., Shima, H. & Fukamichi, K. Magnetization reversal due to vortex nucleation, dis dot arrays. Phys. Rev. B 65, 024414 (2002)

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Cowburn, R. P., Koltsov, D. K., Adeyeye, A. O., Welland, M. E. & Tricker, D. M. Single-domain circular nanomagnets. Phys. R

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Thiele, A. A. Steady-State Motion of Magnetic Domains. Phys. Rev. Lett. 30, 230-2 (1973)

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Huber D. L. Dynamics of spin vortices in two-dimensional planar magnets, Phys. Rev. B 26, 3758

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Guslienko, K. Yu. et al. Eigenfrequencies of vortex state excitations in magnetic submicron-size di

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[18]

Usov, N. A. & L. G. Kurkina, L. G. Magnetodynamics of vortex in thin cylindric platelet. J. Magn. Magn. Mater. 242, 1005-1008 (2002)

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Ivanov, B. A. & Zaspel, C. E. High frequency modes in vortex-state nanomagnets, P Rev. Lett. 94, 027205, (2005)

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Park, J. P., Eames, P., Engebretson, D. M., Berezovsky, J. & Crowell, P. A. Imaging of spin dynamics in closure doma

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Buess, M. et al. Fourier transform imaging of spin vortex eigenmodes. Phys. Rev. Lett. 93, 077207 (2004)

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[22]

Zaspel C.E., Ivanov B.A., Park J.P. & Crowell P.A. Excitations in vortex-state permalloy dots. Phys. Rev. B 50

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Puzic A,. et al. Spatially resolved ferromagnetic resonance: Imaging of ferromagnetic eigenmodes. J. Appl. Phys. 97, 10E70

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Choe, S. B. et al. Vortex core-driven magnetization dynamics. Science 304, 420-422 (2004)

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Novosad, V. et al. Spin excitations of magnetic vortices in ferromagnetic nanodots. Ph Rev. B 6

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Giovannini, L. et al. Spin excitations of nanometric cylindrical dots in vortex and saturated magnetic states. P

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Hillebrands, B. & Ounadjela, K. “Spin Dynamics in Confined Magnetic Structures Topics in Applied Physics, Vol. 83 (Springer, Berlin, 2002). 10 he . Rev. B 43, 5992 (1991). - -100 (2002). 86, 223112 (2005). of 9, 214404 (2004). hys. Rev

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Voelkel A. R., Mertens F. G., Bishop A. R., & Wysin G. M. Motion of vortex pairs in t ferromagnetic and antiferromagnetic Heisenberg model. Phys

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Voelkel A. R., Wysin G. M., Mertens F. G., Bishop A. R., & Schnitzer H. J. Collective- variable approach to the dynamics of nonlinear magnetic excitations with application to vortices. Phys. Rev. B 50, 12711 (1994)

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Kovalev, A. S., Komineas, S. & Mertens, F. G. Scattering of vortex pairs in 2d easy plane ferromagnets. Eur. Phys. J. B 25, 89

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Guslienko, K. Yu., Buchanan, K. S., Bader, S. D. & Novosad, V. Dynamics of coupled vortices in layered magnetic nanodots. Appl. Phys. Lett

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Shibata, J. & Otani, Y. Magnetic v ortex dynamics in a two-dimensional square lattice ferromagnetic nanodisks. Phys. Rev. B 70, 12404 (2004)

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Usov, N. A., Chang, C. R. & Wei, Z. H. Buckling instability in thin soft elliptical particles. Phys. Rev. B 66, 184431 (2002)

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[34]

Vavassori, P. et al. Magnetization reversal via single and double vortex states in submicron Permalloy ellipses. Phys. Rev B 6

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[35]

Novosad, V. et al. Magnetic vortex resonance in patterned ferromagnetic dots. P B 72, 024455 (2005). 11 Figure captions Figure 1: MFM and simulations of the two-vortex re manent states. a) MFM images of a 3 by 1.5 µm ellipse, 40 nm thick. Half-size simulations (1.5 by 0.75 µm ellipse, 20 nm thick) show that the in -plane equilibr ium magnetization distrib...

View on Pith 2005

How the literature uses it

1 arXiv paper cite this in 1 sentence

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it

Cites it as support 1 paper

For this purpose rf excitations have been found to be exceptionally efficient, moving domain walls at far lower current densities than pulsed voltages [ref] .

Electrical Detection of Magnetization Dynamics via Spin Rectification Effects · Electrical Detection of Domain Wall Dynamics · 0.96

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [2]

Manton, N. & Sutcliffe, P. Topological Solitons (Cambridge University Press, C 2004)

View on Pith 2004

[2] [3]

Khaykovich, L. et al. Formation of a matter-wave bright soliton. Science 296, 1290-1293 (2002)

View on Pith 2002

[3] [4]

Strecker, K. E., Partridge, G. B., Truscott, A. G. & Hulet, R. G. Formation and propagat

View on Pith

[4] [5]

Nakazawa, M., Yamada, E. & Kubota, H. Coexistence of self-induced transparen and nolinear Schrodinger soliton. Phys. Rev. Lett. 66, 2625-2628 (1991)

View on Pith 1991

[5] [6]

Matthews, M. R. et al. Vortices in a Bose-Einstein condensate. Phys. Rev. Lett. 83, 2498- 2501 (1999)

View on Pith 1999

[6] [7]

Lamb, H. Hydrodynamics (Dover Publications, New York, 1945)

View on Pith 1945

[7] [8]

Kosevich, A 117 (1990)

View on Pith 1990

[8] [9]

Saitoh, E., Miyajima, H., Yamaoka, T. & G. Tatara, G. Current-induced resonance and mass determ

View on Pith

[9] [10]

Argyle, B. E. Terrenzio, E. & Slonczewski, J. C. Magnetic vortex dynamics using the optical Cotton-Mouton effect. Phys. Rev. Lett. 53, 190-193 (1984)

View on Pith 1984

[10] [11]

Shinjo, T., Okuno, T., Hassdorf, R., Shigeto, K. & Ono, T. Magnetic vortex core observation in circular dots of permalloy. Science 289, 930-932 (200 9 tex placement, and annihilation in submicron ferromagnetic ev. Lett. 83, 1042-1045 (1999). 33 (1982). sks. J. Appl. Phys. 91, 8037-8039 (2002). al hys. in and vortex structures. Phys. Rev. B 67, 020403 (200...

View on Pith 1999

[11] [12]

Wachowiak, A. et al. Direct observation of internal spin structure of magnetic vor cores. Science 298, 577-580 (2002)

View on Pith 2002

[12] [13]

Guslienko, K. Yu., Novosad, V., Otani, Y., Shima, H. & Fukamichi, K. Magnetization reversal due to vortex nucleation, dis dot arrays. Phys. Rev. B 65, 024414 (2002)

View on Pith 2002

[13] [14]

Cowburn, R. P., Koltsov, D. K., Adeyeye, A. O., Welland, M. E. & Tricker, D. M. Single-domain circular nanomagnets. Phys. R

View on Pith

[14] [15]

Thiele, A. A. Steady-State Motion of Magnetic Domains. Phys. Rev. Lett. 30, 230-2 (1973)

View on Pith 1973

[15] [16]

Huber D. L. Dynamics of spin vortices in two-dimensional planar magnets, Phys. Rev. B 26, 3758

View on Pith

[16] [17]

Guslienko, K. Yu. et al. Eigenfrequencies of vortex state excitations in magnetic submicron-size di

View on Pith

[17] [18]

Usov, N. A. & L. G. Kurkina, L. G. Magnetodynamics of vortex in thin cylindric platelet. J. Magn. Magn. Mater. 242, 1005-1008 (2002)

View on Pith 2002

[18] [19]

Ivanov, B. A. & Zaspel, C. E. High frequency modes in vortex-state nanomagnets, P Rev. Lett. 94, 027205, (2005)

View on Pith 2005

[19] [20]

Park, J. P., Eames, P., Engebretson, D. M., Berezovsky, J. & Crowell, P. A. Imaging of spin dynamics in closure doma

View on Pith

[20] [21]

Buess, M. et al. Fourier transform imaging of spin vortex eigenmodes. Phys. Rev. Lett. 93, 077207 (2004)

View on Pith 2004

[21] [22]

Zaspel C.E., Ivanov B.A., Park J.P. & Crowell P.A. Excitations in vortex-state permalloy dots. Phys. Rev. B 50

View on Pith

[22] [23]

Puzic A,. et al. Spatially resolved ferromagnetic resonance: Imaging of ferromagnetic eigenmodes. J. Appl. Phys. 97, 10E70

View on Pith

[23] [24]

Choe, S. B. et al. Vortex core-driven magnetization dynamics. Science 304, 420-422 (2004)

View on Pith 2004

[24] [25]

Novosad, V. et al. Spin excitations of magnetic vortices in ferromagnetic nanodots. Ph Rev. B 6

View on Pith

[25] [26]

Giovannini, L. et al. Spin excitations of nanometric cylindrical dots in vortex and saturated magnetic states. P

View on Pith

[26] [27]

Hillebrands, B. & Ounadjela, K. “Spin Dynamics in Confined Magnetic Structures Topics in Applied Physics, Vol. 83 (Springer, Berlin, 2002). 10 he . Rev. B 43, 5992 (1991). - -100 (2002). 86, 223112 (2005). of 9, 214404 (2004). hys. Rev

View on Pith 1991

[27] [28]

Voelkel A. R., Mertens F. G., Bishop A. R., & Wysin G. M. Motion of vortex pairs in t ferromagnetic and antiferromagnetic Heisenberg model. Phys

View on Pith

[28] [29]

Voelkel A. R., Wysin G. M., Mertens F. G., Bishop A. R., & Schnitzer H. J. Collective- variable approach to the dynamics of nonlinear magnetic excitations with application to vortices. Phys. Rev. B 50, 12711 (1994)

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